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News from the world of maths: Sad news: Paul Cohen dies

Sad news: Paul Cohen dies

The distinguished mathematician Paul Cohen sadly died on Friday the 23rd of March 2007, just a few days before his 73rd birthday.

Cohen worked on a range of topics, but is best-known for his work on set theory. His work in this area cuts right down to the foundations of mathematics. The mathematician David Hilbert famously believed that it should be possible to phrase all of mathematics in a single and completely formal theory.
Based on a collection of axioms — pre-determined facts that are so self-evident they do not themselves need to be proved — and the rules of logic, it should be possible to formally prove every mathematical truth and to arrive at a complete theory that is free from contradiction.

Set theory provides a language in which such a formal system might be phrased. In maths a set is simply a collection of objects. The objects themselves are allowed to remain abstract and so you can talk about all mathematical objects — whether they are functions, numbers or anything else — in terms of sets. In the beginning of the twentieth century, mathematicians laid down a list of axioms
and rules of logic that resulted in a formal and rigorous theory of sets. The system is known as ZF theory, after Ernst Zermelo and Abraham Fraenkel.

On the face of it, sets are simple objects. But once you allow them to have an infinite number of elements, things become complicated. If you take the set of whole numbers, for example, and compare it to the set of all numbers, you'll notice that although both sets are infinite, they are fundamentally different from each other. The set of whole numbers consists of isolated objects,
whereas you can think of all the numbers as merging together to give a continuum. In some sense, the set of all numbers is "bigger" than the set of whole numbers, so we need a notion of size for infinite sets also. The mathematician Georg Cantor (1845 - 1918) formalised such a notion, called cardinality, and
in doing so came up with a conjecture that became known as the continuum hypothesis: that there is no set that is "larger" than the set of whole numbers and "smaller" than the set of all numbers. However, a proof of this fact illuded Cantor and the continuum hypothesis became the first on Hilbert's list of mathematical challenges for the 20th century.

When it comes to the axioms of set theory, things aren't all that clear-cut either. Some axioms are clear as day, for example the one which states that two sets are the same if all their elements are the same. Others are more controversial though, and one of them is the axiom of choice. It states that if you have a collection of sets, then you can form a new set by picking one element
from each. Again, this is clearly possible when you've got a finite collection of sets, but if there are infinitely many, it's not clear that a mechanism for picking an element from each always exists. Although the majority of mathematicians accept the axiom of choice, there is a school of thought which doubts it.

Paul Cohen proved two significant — and to the Hilbert school of thought disappointing — results in this area. He showed that neither the continuum hypothesis nor the axiom of choice can be proved from the axioms of ZF theory. Together with results previously proved by Kurt Gödel, this means that neither can be proved to be either true or false within ZF theory. Within ZF theory, the continuum
hypothesis will forever remain a mystery. As far as the axiom of choice is concerned, the result means that there's no clear indication as to whether you should include it or its negation as one of your initial axioms of set theory. In both cases you come up with a sound system, so the decision whether to include it or not has to be made on different grounds.

In proving these results, Cohen not only contributed to the philosophical debate about the foundations of maths, but also developed a whole new set of tools to deal with questions on what can and cannot be proved within formal mathematical systems. His work was honoured with a Fields medal in 1966. The world of maths has lost one of its most distinguished members.